And here is the solution:

$396 = 2^2\cdot 3^2\cdot 11$, so one of $A$ or $B$ is divisible by $11$. Since $\gcd(B,C) = 33 = 3\cdots 11$, it must be the case that $B$ is divisible by $11$.

This implies that $A$ is not divisible by $11$. Further, since $\gcd(A,B) = 12 = 2^2\cdot 3$ and $\gcd(B,C) = 3\cdot 11$, we know that $C$ is not divisible by $2$. Hence, $\gcd(A,C)$ is a power of $3$, so either $3$ or $9$ (both of which are possible).